Leveraging the spectral decomposition of \(\mathbf {S}\) , graph filters and signals can be represented in the frequency domain.
To be precise, let us use the eigenvectors of \(\mathbf {S}\)  to define the \(\text{N}\times \text{N}\)  matrix \(\mathbf {U}:=\mathbf {V}^{-1}\) , and the eigenvalues of \(\mathbf {S}\)  to define the \(\text{N}\times \text{L}\)  Vandermonde matrix \(\mathbf {\Psi }\) , where \(\Psi _{ij}:=(\Lambda _{ii})^{j-1}\) .
Using these conventions, the frequency representations of a signal \(\mathbf {x}\)  and of a filter \(\mathbf {h}\)  are defined as \(\widehat{\mathbf {x}}:=\mathbf {U}\mathbf {x}\)  and \(\widehat{\mathbf {h}}:=\mathbf {\Psi }\mathbf {h}\) , respectively [1]}, [2]}.
Exploiting such representations, the output \(\mathbf {y}\!=\!\mathbf {H}\mathbf {x}\)  of a graph filter in the frequency domain is given by
\(\widehat{\mathbf {y}}=\operatorname{diag}\big (\mathbf {\Psi }\mathbf {h}\big )\mathbf {U}\mathbf {x}=\operatorname{diag}\big (\widehat{\mathbf {h}}\big )\widehat{\mathbf {x}}=\widehat{\mathbf {h}}\circ \widehat{\mathbf {x}},\) 
